Till innehåll på sidan

# Victor Turchin: Smoothing theory deloopings of disk embedding and diffeomorphism spaces

Tid: To 2022-01-13 kl 14.15 - 16.00

Plats: Zoom, meeting ID: 921 756 1880

Videolänk: https://kva-se.zoom.us/j/9217561880

Föreläsare: Victor Turchin (Kansas State University)

Abstract: The smoothing theory provides delooping to the groups of relative to the boundary disk diffeomorphisms $$\mathrm{Diff}_\partial(D^m) = \Omega^{m+1} \mathrm{Top}_m/\mathrm{O}_m, m\neq 4$$; $$\mathrm{Diff}_\partial(D^m) = \Omega^{m+1} \mathrm{PL}_m/\mathrm{O}_m, \textrm{any } m$$. This result was established in the 70s and is due to contributions of several people: Cerf, Morlet, Burghelea, Lashof, Kirby, Siebenmann, Rourke, etc. In the talk I will briefly explain how this result is obtained. Less known is a similar statement for the spaces $$\mathrm{Emd}_\partial(D^m,D^n), \mathrm{Emd}^{fr}_\partial(D^m,D^n)$$ of relative to the boundary (framed) disk embedding spaces. This latter result was hidden in a work of Lashof from 70s and was stated explicitly by Sakai nine years ago. The range stated by Sakai is $$n>4, n-m>2$$. However, after a careful reading of the literature and with the help of Sander Kupers we got convinced that the delooping in question holds for any codimension $$n-m$$ and any $$n$$ (except $$n=4$$ in the topological version of delooping). Of particular interest is the case $$m=2, n=4$$. In the talk I will also explain how the smoothing theory techniques can be used to show that the delooping is compatible with the Budney $$E_{m+1}$$-action. The starting point in this project was the question whether it is possible to combine the Budney $$E_{m+1}$$ action on $$\mathrm{Diff}_\partial(D^m)$$ and $$\mathrm{Emb}^{fr}_\partial(D^m,D^n)$$ with Hatcher's $$\mathrm{O}_{m+1}$$ action on these spaces into an $$\mathrm{E}_{m+1}^{fr}$$-action. The answer is yes, it can be done by means of the smoothing theory delooping. Joint project in progress with Paolo Salvatore.